when the probability of an event decreases.

Question

we have 5 mutually exclusive events

we know the probability of each event.

for some reason the probability of one event has become half the previous one, how to find the new probability of each event.

Thank you.

Answer 1

Let's say the events are A,B,C,D,E, and let's assume the probabilities of these events occurring are a,b,c,d,e respectively. Now let's assume the probability of event A goes does down to a/2.

Now, the probabilities still need to add up to 1, so b,c,d,e should increase for a total of a/2. So, let's make one more assumption, which is that the probabilities increase proportionally to their original probabilities (e.g. if the probabilitity of B was twice the probability of C, then afterwards we still want the new probability B to be twice the new probability of C). With that assumption, the new probabilities would be:

$b_{new} = b + \frac{b}{b+c+d+e}*\frac{a}{2}$

$c_{new} = c + \frac{c}{b+c+d+e}*\frac{a}{2}$

$d_{new} = d + \frac{d}{b+c+d+e}*\frac{a}{2}$

$e_{new} = e + \frac{e}{b+c+d+e}*\frac{a}{2}$

As an example: if a goes down from 40% to 20%, then the others should go up by a total of 20%. But assuming that b,c,d, and e all go up by an equal 5% would mean that b ends up at 6% and c at 45%, meaning that C would go from being 40 times as likely as B to merely being 7.5 times as likely. Of course, we don't really know how the change in a is going to effect the changes in the other probabilities, but the way probabilities come about, one might expect this ratio between probabilities to stay somewhat the same. For example, suppose we flip a coin, and if it comes up heads, that's event A. But if it comes up tails, we do something else (e.g. throw dice, pick marbles from urns, or do whatever self-respecting probability theorist seems to enjoy doing)... and depending on the possible outcomes we get there, we get events B,C,D, and E. But now suppose the coin becomes biased, and the probability of A is now halved. Well, if the other events are not effected by this bias, then the probabilities of b,c,d, and e will all still be proportional to each other, and thus you would get the equations from above.

Of course, what if b,c,d,e do get effected by the bias of the coin? E.g.

Heads -> A

Tails, Heads -> B

Tails, Tails, Heads -> C

Tails, Tails, Tails, Heads -> D

Tails, Tails, Tails, Tails -> E

Now the coin becoming biased (P(Heads) = 1/4) has the following result:

a was 1/2, but goes down to 1/4

b was 1/4, and becomes 3/16 (so goes down a bit as well)

c was 1/8, and becomes 9/64 (so goes up a tiny bit)

d was 1/16 and becomes 27/256 (so goes up, and more so than c)

e was 1/16 and becomes 81/256 (so goes up a lot!)

So now you get completely different results!

In sum, then, without further information, we really can't establish how b,c,d, and e are effected by a going down ... but we can if we make a somewhat reasonable assumption.